Abstract

Let 𝐺 be a connected graph on 𝑉. A subset 𝑋 of 𝑉 is all-paths convex (or ap-convex) if 𝑋 contains each vertex on every path joining two vertices in 𝑋 and is monophonically convex (or 𝑚-convex) if 𝑋 contains each vertex on every chordless path joining two vertices in 𝑋. First of all, we prove that ap-convexity and 𝑚-convexity coincide in 𝐺 if and only if 𝐺 is a tree. Next, in order to generalize this result to a connected hypergraph 𝐻, in addition to the hypergraph versions of ap-convexity and 𝑚-convexity, we consider canonical convexity (or 𝑐-convexity) and simple-path convexity (or sp-convexity) for which it is well known that 𝑚-convexity is finer than both 𝑐-convexity and sp-convexity and sp-convexity is finer than ap-convexity. After proving sp-convexity is coarser than 𝑐-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of 𝛾-acyclic hypergraphs.

1. Introduction

Convexity is a fundamental concept occurring in geometry, topology, and functional analysis, and the problem of computing convex hulls is at the core of many computer engineering applications, for instance, in robotics, computer graphics, or optimization (see page 125 in [1]). For the theory of abstract convex structures see [2].

This paper is motivated by the paper by Farber and Jamison [3] where structural properties of different notions of convexity in graphs and hypergraphs are stated. We focus on the following four notions of convexities in hypergraphs: 𝑚-convexity (for monophonic convexity) [4], 𝑐-convexity (for canonical convexity) [4], sp-convexity (for simple-path convexity) [3], and ap-convexity (for all-paths convexity), which are known to be related to each other by the following implications:ap-convex⟹sp-convex⟹𝑚-convex𝑐-convex⟹𝑚-convex.(1.1)
Note that hypergraph 𝑚-convexity generalizes graph 𝑚-convexity [3, 5], and both hypergraph sp-convexity and ap-convexity generalize graph ap-convexity [6, 7]. After proving the implication sp-convex ⇒𝑐-convex, we characterize the hypergraphs in which each pair of the four convexities above is equivalent.

As an application, suppose that for two hypergraph convexities 𝜃 and 𝜃 we have two algorithms 𝐴 and 𝐴 to compute 𝜃-convex hulls and 𝜃-convex hulls for hypergraphs and that the computational complexity of 𝐴 is less than 𝐴. If 𝜃 and 𝜃 are equivalent in a class of hypergraphs then, for every hypergraph 𝐻 belonging to this class, we can compute 𝜃-convex hulls using the algorithm 𝐴 instead of 𝐴.

As a consequence of our equivalence results, we obtain the following convexity-theoretic characterizations of the so-called “Berge-acyclic” hypergraphs and “𝛾-acyclic” hypergraphs [8]:(i)a hypergraph is Berge-acyclic if and only if 𝑐-convexity and ap-convexity are equivalent in the hypergraph (see Theorem 6.9);(ii)a hypergraph is Berge-acyclic if and only if 𝑚-convexity and ap-convexity are equivalent in the hypergraph (see Theorem 6.10);(iii)a hypergraph is 𝛾-acyclic if and only if 𝑐-convexity and sp-convexity are equivalent in the hypergraph (see Theorem 5.5);(iv)a hypergraph is 𝛾-acyclic if and only if 𝑚-convexity and sp-convexity are equivalent in the hypergraph (see Theorem 5.6).

It is worth noting that Berge-acyclic and 𝛾-acyclic hypergraphs belong to a family of hypergraphs (𝛼-,𝛽-,𝛾-, and Berge-acyclic hypergraphs) which enjoy a number of theoretical and computational properties in database theory [8, 9], artificial intelligence [10, 11], and statistics [12, 13]. Moreover, 𝑐-convexity is a “convex geometry” in 𝛼-acyclic hypergraphs [4], and 𝛽-acyclicity characterizes those hypergraphs in which sp-convexity yields a “convex geometry” [3].

To achieve the above-mentioned results, we need several standard hypergraph-theoretic definitions (such as the four acyclicity types above) and additional notions which are introduced ad hoc. The outline of the paper is as follows. Section 2 contains basic definitions and results on graphs and hypergraphs. In Section 3 we review basic results on structural properties of 𝑚-convexity, 𝑐-convexity, sp-convexity, and ap-convexity in graphs and hypergraphs; moreover, we state some preliminary results. In Section 4 we characterize the hypergraphs in which 𝑚-convexity and 𝑐-convexity are equivalent. In Section 5 we prove that 𝛾-acyclicity characterizes those hypergraphs in which 𝑐-convexity (or 𝑚-convexity) and sp-convexity are equivalent. In Section 6 we characterize the hypergraphs in which sp-convexity and ap-convexity are equivalent; moreover, we prove that Berge-acyclicity characterizes those hypergraphs in which 𝑐-convexity (or 𝑚-convexity) and ap-convexity are equivalent. Section 7 contains an open problem for future research.

2. Terminology and Notation

2.1. Graphs

Henceforth, we only consider graphs [14] with no loops and no multiple edges, which henceforth will be referred simply to as graphs. Let 𝐺 be a graph. Two vertices of 𝐺 are adjacent if they are joined by some edge of 𝐺. A nonempty subset 𝑋 of 𝑉(𝐺) is a clique if every two distinct vertices in 𝑋 are adjacent. The subgraph of 𝐺 induced by a nonempty subset 𝑋 of 𝑉(𝐺) is the graph, denoted by 𝐺[𝑋], with vertex set 𝑋 in which two distinct vertices are adjacent if and only if they are adjacent in 𝐺. The notation 𝐺[𝑉⧵𝑋] is abridged into 𝐺⧵𝑋.

A path is a sequence 𝑝=(𝑎0,𝑎1,…,𝑎𝑘), 𝑘≥1, of distinct vertices such that 𝑎𝑖−1 and 𝑎𝑖 are adjacent for 1≤𝑖≤𝑘. The path 𝑝 is said to join 𝑎0 and 𝑎𝑘 (or, equivalently, to be an 𝑎0–𝑎𝑘 path) and to have length 𝑘; moreover, if 𝑘>1, 𝑝 is said to pass through each 𝑎ℎ, 1≤ℎ≤𝑘−1, and two vertices 𝑎𝑖 and 𝑎𝑗 on 𝑝 are said to be consecutive if |𝑖−𝑗|=1. By 𝑉(𝑝) we denote the vertex set {𝑎0,𝑎1,…,𝑎𝑘}.

The distance between two vertices 𝑢 and 𝑣 is the length of any minimum-length 𝑢–𝑣 path. A graph 𝐺 is distance hereditary if, for every two vertices 𝑢 and 𝑣 of 𝐺 and for every connected, induced subgraph 𝐺 containing 𝑢 and 𝑣, the distances between 𝑢 and 𝑣 in 𝐺 and 𝐺 are the same.

A cycle of length 𝑘, 𝑘≥2, is a sequence 𝑐=(𝑎0,𝑎1,…,𝑎𝑘−1,𝑎0) where (𝑎0,𝑎1,…,𝑎𝑘−1) is a path, and the vertices 𝑎0 and 𝑎𝑘−1 are adjacent. Two vertices 𝑎𝑖 and 𝑎𝑗 on the cycle 𝑐 are consecutive if either |𝑖−𝑗|=1 or |𝑖−𝑗|=𝑘−1. By 𝑉(𝑐) we denote the set of vertices {𝑎0,𝑎1,…,𝑎𝑘−1}. A chord of 𝑐 is an edge joining two nonconsecutive vertices on 𝑐.

A graph is chordal if every cycle of length at least 4 has a chord. A graph is strongly chordal if it is chordal and, for every cycle 𝑐 of even length, there are two nonconsecutive vertices at odd distance on 𝑐 that are adjacent. A graph is Ptolemaic if it is distance hereditary and chordal.

Let 𝐺 be a graph with at least two vertices. A vertex 𝑢 of 𝐺 is a cut vertex (or an “articulation point”) of 𝐺 if the number of connected components of 𝐺⧵{𝑢} is greater than the number of connected components of 𝐺 or, equivalently, there exist two vertices 𝑣≠𝑢 and 𝑤≠𝑢 in the connected component of 𝐺 containing 𝑢 such that every 𝑣–𝑤 path passes through 𝑢 [14]. A block of 𝐺 is a maximal connected partial graph of 𝐺 containing no cut vertices. A block of 𝐺 is trivial if it consists of two vertices and nontrivial otherwise. Finally, 𝐺 is a block graph if the vertex set of every block of 𝐺 is a clique.

Proposition 2.1 (see [14]). Let 𝐺 be a nontrivial block. For every three distinct vertices 𝑢,𝑣, and 𝑤 of 𝐺, there exists a 𝑣–𝑤 path that passes through 𝑢.

Proposition 2.2 (see [14]). Let 𝐺 be a connected graph, and let 𝐵 be a block of 𝐺. If 𝑤 is not a vertex of 𝐵, then 𝐵 contains a cut vertex 𝑢 of 𝐺 such that, for every vertex 𝑣≠𝑢 of B, every 𝑣–𝑤 path passes through 𝑢.

Let 𝐺 be a graph. Two connected vertices are separated by a subset 𝑆 of 𝑉(𝐺) if they are in two distinct components of the induced graph 𝐺⧵𝑋. A nonempty subset 𝑋 of 𝑉(𝐺) is nonseparable if 𝐺[𝑋] is connected and no two vertices in 𝑋 are separated by a clique of 𝐺. The prime components of 𝐺 are the subgraphs of 𝐺 induced by maximal nonseparable sets.

2.2. Hypergraphs

Generalizing notions and convexities from graphs to hypergraphs is not always straightforward, because there are often several nonequivalent ways to do this and different terminologies. This is true also for notions that hypergraph convexities are based on. For example, “simple paths” in [3] are called “chordless chains” in [15], “simple circuits” in [3] are called “chordless cycles” in [15], and “weak 𝛽-cycles” in [3] if they are of length at least 3, “nest” vertices in [3] are called “simple” vertices in [15].

A (generic) hypergraph is a (possibly empty) set 𝐻 of nonempty sets; the elements of 𝐻 are the (hyper) edges of 𝐻 and their union is the vertex set of 𝐻, denoted by 𝑉(𝐻).

A hypergraph is trivial if it has only one edge and nontrivial otherwise. A partial (sub) hypergraph of hypergraph 𝐻 is any subset of 𝐻. A hypergraph is simple if no edge is contained in another edge. The reduction of a hypergraph 𝐻 is the partial hypergraph of 𝐻 whose edges are the maximal (with respect to set-inclusion) edges of 𝐻.

Let 𝑋 be a nonempty subset of 𝑉(𝐻). The subhypergraph of 𝐻induced by 𝑋 is the hypergraph, denoted by 𝐻[𝑋], whose edges are exactly the maximal (with respect to set-inclusion) edges of the hypergraph {𝐴∩𝑋∶𝐴∈𝐻}⧵{∅}. The notation 𝐻[𝑉(𝐻)⧵𝑋] is abridged into 𝐻⧵𝑋.

A partial edge is a nonempty vertex set that is contained in some edge. Two vertices are adjacent if they belong together in some edge. A nonempty subset 𝑋 of 𝑉(𝐻) is a clique if every two distinct vertices in 𝑋 are adjacent. A hypergraph is conformal if every clique is a partial edge.

A path is a sequence 𝑝=(𝑎0,𝐴1,𝑎1,…,𝐴𝑘,𝑎𝑘), 𝑘≥1, where the 𝑎𝑖’s are pairwise distinct vertices, the 𝐴𝑖’s are pairwise distinct edges, and {𝑎𝑖−1,𝑎𝑖}⊆𝐴𝑖 for 1≤𝑖≤𝑘. The path 𝑝 is said to join 𝑎0 and 𝑎𝑘 (or, equivalently, to be an 𝑎0–𝑎𝑘 path), to have length 𝑘 and, if 𝑘>1, to pass through each 𝑎ℎ, 1≤ℎ≤𝑘−1. Moreover, two vertices 𝑎𝑖 and 𝑎𝑗 on the path 𝑝 are consecutive if |𝑖−𝑗|=1. Finally, by 𝑉(𝑝) we denote the vertex set {𝑎0,𝑎1,…,𝑎𝑘}, and by 𝐻(𝑝) we denote the partial hypergraph {𝐴1,…,𝐴𝑘−1,𝐴𝑘} of 𝐻.

A hypergraph is connected if any two vertices are joined by a path. A nonempty subset 𝑋 of 𝑉(𝐻) is connected if the induced subhypergraph 𝐻[𝑋] is connected. Note that if 𝐻 is a graph, then the subgraph of 𝐻 induced by 𝑋 equals the reduction of 𝐻[𝑋].

The connected components of 𝐻 are the subhypergraphs of 𝐻 induced by maximal connected subsets of 𝑉(𝐻).

The following definitions of an “articulation set” and of a “block” in a hypergraph are the natural generalizations of the notions of a cut vertex and of a block in a graph [8, 17].

Let 𝐻 be a reduced hypergraph. A separator is a nonempty subset 𝑆 of 𝑉(𝐻) such that there exist two connected vertices of 𝐻 that are in two distinct components of the induced subhypergraph 𝐻⧵𝑆. A separator of 𝐻 is an articulation set if it is the intersection of two edges of 𝐻. A nonempty subset 𝑋 of 𝑉(𝐻) is nonseparable if 𝐻[𝑋] is connected and has no articulation set. A block of 𝐻 is the reduction of the subhypergraph of 𝐻 induced by a maximal nonseparable set.

Example 2.3. Let 𝐻={𝐴1,𝐴2,𝐴3,𝐴4,𝐴5}, where 𝐴1={1,2}, 𝐴2={1,3}, 𝐴3={2,4}, 𝐴4={3,4,5}, and 𝐴5={4,5,6}. The hypergraph 𝐻 is shown in Figure 1. The set {4,5} is the only articulation set of 𝐻, and the blocks of 𝐻 are shown in Figure 2.

Figure 1

Figure 2

Finally, with a hypergraph 𝐻 we can associate two graphs: the “2-section” of 𝐻 and the “incidence graph” of 𝐻, which are defined as follows.

The 2-section (also called “adjacency graph” or “underlying graph” or “primal graph”, or “Gaifman graph”) of 𝐻 is the graph with vertex set 𝑉(𝐻) in which two vertices are adjacent if and only if they are adjacent in 𝐻. We denote the 2-section of 𝐻 by 𝐻[2].

The incidence graph of 𝐻 is the bipartite graph with bipartition (𝑉(𝐻),𝐻), where 𝑎∈𝑉(𝐻) and 𝐴∈𝐻 are joined by an edge if and only if 𝑎∈𝐴. We denote the incidence graph of 𝐻 by 𝐺(𝐻), and the size of 𝐻 is the number of vertices and edges of 𝐺(𝐻) [18]. Note that, if 𝐻 is connected, then the size of 𝐻 is 𝑂(𝑒) where 𝑒 is the number of edges of 𝐺(𝐻).

2.3. Acyclicity

Fagin [8] introduced four notions of acyclicity for hypergraphs which are now recalled and, in the next sections, will be proven to be closely related to hypergraph convexities.

A cycle (also called a “circuit” [3]) is a sequence 𝑐=(𝑎0,𝐴1,𝑎1,…,𝐴𝑘−1,𝑎𝑘−1,𝐴𝑘,𝑎0), 𝑘≥2, where (𝑎0,𝐴1,𝑎1,…,𝐴𝑘−1,𝑎𝑘−1) is a path, {𝑎0,𝑎𝑘−1}⊆𝐴𝑘 and 𝐴𝑘≠𝐴ℎ for 1≤ℎ≤𝑘−1. The cycle 𝑐 is said to have length 𝑘; moreover, two vertices 𝑎𝑖 and 𝑎𝑗 on 𝑐 are consecutive if either |𝑖−𝑗|=1 or |𝑖−𝑗|=𝑘−1. By 𝑉(𝑐) we denote the set of vertices {𝑎0,𝑎1,…,𝑎𝑘−1}, and by 𝐻(𝑐) we denote the partial hypergraph {𝐴1,…,𝐴𝑘−1,𝐴𝑘} of 𝐻.

A 𝛾-cycle is a cycle 𝑐 of length at least 3 such that at most one vertex in 𝑉(𝑐) belongs to three or more edges of 𝐻(𝑐).

A 𝛽-cycle (a “weak 𝛽-cycle” in [8]) is a cycle 𝑐 of length at least 3 such that every vertex in 𝑉(𝑐) belongs to exactly two edges of 𝐻(𝑐).

A hypergraph is Berge-acyclic if it contains no cycles, 𝛾-acyclic if it contains no 𝛾-cycles, and 𝛽-acyclic if it contains no 𝛽-cycles (or, equivalently, if every partial hypergraph is 𝛼-acyclic).

A reduced hypergraph is 𝛼-acyclic if all its blocks are trivial hypergraphs. A hypergraph is 𝛼-acyclic precisely if its reduction is 𝛼-acyclic.

It is well known [8] that the following implications on hypergraphs hold:Berge-acyclic⟹𝛾-acyclic⟹𝛽-acyclic⟹𝛼-acyclic(2.1)
but none of their reverse implications holds in general.

Several characterizations of Berge-acyclicity, 𝛾-acyclicity, 𝛽-acyclicity, and 𝛼-acyclicity exist, and the following is based on the 2-section of a hypergraph.

Proposition 2.4 (see [19]). A hypergraph is Berge-acyclic if and only if it is conformal and its 2-section is a block graph. A hypergraph is 𝛾-acyclic if and only if it is conformal and its 2-section is a Ptolemaic graph. A hypergraph is 𝛽-acyclic if and only if it is conformal and its 2-section is a strongly chordal graph. A hypergraph is 𝛼-acyclic if and only if it is conformal and its 2-section is a chordal graph.

By Proposition 2.4, 𝛽-acyclic hypergraphs are the same as “totally balanced” hypergraphs [3] and as “totally decomposable” hypergraphs in [12]. Finally, note that 𝛼-acyclic hypergraphs are called “acyclic” hypergraphs in [4, 9, 11, 17, 18] and “decomposable” hypergraphs in [12, 13, 20, 21].

Before closing this subsection, we mention two 𝛼-acyclic hypergraphs which in some sense represent the “superstructure” of a graph and of a hypergraph.

The prime hypergraph of graph 𝐺 is the (reduced) hypergraph whose edges are precisely the vertex sets of the prime components of 𝐺.

Proposition 2.5 (see [21]). The prime hypergraph of a graph is a reduced, 𝛼-acyclic hypergraph.

A nonempty subset 𝑋 of 𝑉(𝐻) is a compact set of 𝐻 if 𝐻[𝑋] is connected and no partial edge of 𝐻[𝑋] is a separator. Note that if 𝑋 is a compact set, then 𝐻[𝑋] has no articulation set; but the reverse does not hold in general (see Example 2.3). A compact component of 𝐻 is the reduction of the subhypergraph of 𝐻 induced by a maximal compact set. The compact hypergraph of 𝐻 is the (reduced) hypergraph whose edges are precisely the vertex sets of the compact components of 𝐻.

Proposition 2.6 (see [22, 23]). The compact hypergraph of a hypergraph is a reduced, 𝛼-acyclic hypergraph. Moreover, a hypergraph is 𝛼-acyclic if and only if its reduction equals its compact hypergraph.

Example 2.3 (continued). The compact components of 𝐻 are shown in Figure 3 and the compact hypergraph of 𝐻 is {{1,2,3,4},{3,4,5},{4,5,6}}.

Figure 3

3. Convexities in Graphs and Hypergraphs

In this section, we recall the definitions and basic results on some convexities in graphs and hypergraphs. Moreover, we state some preliminary results.

Let 𝐻 be a connected hypergraph. A set 𝜃(𝐻) of subsets of 𝑉(𝐻) is a convexity space [24, 25] if(i)the empty set, the singletons, and 𝑉(𝐻) belong to 𝜃(𝐻),(ii)𝜃(𝐻) is closed under set intersection,(iii)every set in 𝜃(𝐻) is connected.

The sets in a convexity space 𝜃(𝐻) are called the 𝜃-convex sets of 𝐻. For a subset 𝑋 of 𝑉(𝐻), the 𝜃-convex hull of 𝑋 is the minimal (with respect to set inclusion) 𝜃-convex set of 𝐻 that includes 𝑋.

3.1. Graph Convexities

In this section, we recall the definitions and basic results on monophonic convexity (𝑚-convexity) and all-paths convexity (ap-convexity) on a connected graph [14, 17, 24–26]. Let 𝐺 be a connected graph. By 𝑚(𝐺) and ap(𝐺) we denote the 𝑚-convexity space and the ap-convexity space on 𝐺, respectively.

A chord of a path 𝑝 is an edge of 𝐺 that joins two nonconsecutive vertices on 𝑝. A path is chordless (or “induced” or “minimal”) if it has no chords. A subset 𝑋 of 𝑉(𝐺) is 𝑚-convex if, for every chordless path 𝑝 joining two vertices in 𝑋, each vertex on 𝑝 belongs to 𝑋.

The following result provides a known characterization of 𝑚-convex sets. Let 𝑋 be a subset of 𝑉(𝐻); an 𝑋–𝑋 path is a path 𝑝=(𝑎0,𝐴1,𝑎1,…,𝐴𝑘,𝑎𝑘) with 𝑉(𝑝)∩𝑋={𝑎0, 𝑎𝑘}.

Theorem 3.1 (see [5]). Let 𝐺 be a connected graph. A subset 𝑋 of 𝑉(𝐺) is 𝑚-convex if and only if, for every two distinct vertices 𝑢 and 𝑣 in 𝑋, 𝑢 and 𝑣 are joined by an 𝑋–𝑋 path, then 𝑢 and 𝑣 are adjacent in 𝐺.

Let 𝐺 be a connected graph with 𝑛 vertices and 𝑚 edges. Dourado et al. [25] gave an algorithm for computing the 𝑚-convex hull of a subset 𝑋 of 𝑉(𝐺) which runs in 𝑂(𝑚𝑛2|𝑋|2) time. A better algorithm was given by Kannan and Changat [27], which runs in 𝑂(𝑚𝑛) time.

A subset 𝑋 of 𝑉(𝐺) is ap-𝑐𝑜𝑛𝑣𝑒𝑥 if, for every path 𝑝 joining two vertices in 𝑋, each vertex on 𝑝 belongs to 𝑋. It is easily seen that if 𝐺 is a tree, then the ap-convex hull of a subset 𝑋 of 𝑉(𝐺) can be computed in 𝑂(𝑚) time simply by deleting the leaves of 𝐺 that are not in 𝑋.

We now state a characterization of those graphs 𝐺 on which ap(𝐺) = 𝑚(𝐺).

Theorem 3.2. Let 𝐺 be a connected graph.The equality ap(𝐺)=𝑚(𝐺) holds if and only if 𝐺 is a tree.

Proof. Since every chordless path is trivially a path, the inclusion ap(𝐺)⊆𝑚(𝐺) is obvious. If 𝐺 is a tree, then trivially one has ap(𝐺)=𝑚(𝐺). Assume that 𝐺 is not a tree. To prove that ap(𝐺)≠𝑚(𝐺), consider any nontrivial block of 𝐺, say 𝐵. Let 𝑢 and 𝑣 be two adjacent vertices of 𝐵. The path (𝑢,𝑣) is the only chordless 𝑢–𝑣 path in 𝐺 and, hence, {𝑢,𝑣} is 𝑚-convex. On the other hand, {𝑢,𝑣} is not ap-convex since, by Propositions 2.1 and 2.2, its ap-convex hull is 𝑉(𝐵), which proves that ap(𝐺)≠𝑚(𝐺).

By Theorem 3.2, if 𝐺 is a tree, then the 𝑚-convex hull of a subset 𝑋 of 𝑉(𝐺) coincides with the ap-convex hull of 𝑋 and, hence, can be computed in 𝑂(𝑚) time.

3.2. Hypergraph Convexities

Let 𝐻 be a connected hypergraph. In this section we recall the definitions and basic results on monophonic convexity (𝑚-convexity), canonical convexity (𝑐-convexity), simple-path convexity (sp-convexity), and all-paths convexity (ap-convexity) on 𝐻. By 𝑚(𝐻), 𝑐(𝐻), sp(𝐻), and ap(𝐻) we denote the 𝑚-convexity space, the 𝑐-convexity space, the sp-convexity space, and the ap-convexity space on 𝐻, respectively.

3.2.1. 𝑚-Convexity

Hypergraph 𝑚-convexity [4] was not defined in terms of paths but using the hypergraph-theoretic version of Theorem 3.1, that is, a subset 𝑋 of 𝑉(𝐻) is 𝑚-convex if, for every two distinct vertices 𝑢 and 𝑣 in 𝑋, 𝑢 and 𝑣 are joined by an 𝑋–𝑋 path, then 𝑢 and 𝑣 are adjacent in 𝐻.

We now recall a useful characterization of 𝑚-convex sets. To this end, we need further definitions.

Let 𝑋 be a subset of 𝑉(𝐻); two edges 𝐴 and 𝐵 of 𝐻 are connected outside 𝑋, written 𝐴≡𝑋𝐵, if(i)𝐴=𝐵 or(ii)(𝐴∩𝐵)⧵𝑋≠∅ or(iii)there exists an edge 𝐶 of 𝐻 such that (𝐴∩𝐶)⧵𝑋≠∅ and 𝐵≡𝑋𝐶.

The edge relation ≡𝑋 is an equivalence relation; the classes of the resultant partition of 𝐻 will be referred to as the 𝑋-components of 𝐻. The hypergraph 𝐻 is 𝑋-connected if 𝐻 has exactly one 𝑋-component. Note that 𝐻 is 𝑋-connected if the graph 𝐻[2]⧵𝑋 is connected and no edge of 𝐻 is completely contained in 𝑋. For an 𝑋-component 𝐾 of 𝐻, we call the set 𝑋∩𝑉(𝐾) the boundary of 𝑋 with 𝐾.

Example 3.3. Consider again the hypergraph 𝐻={𝐴1,𝐴2,𝐴3,𝐴4,𝐴5} of Example 2.3 (see Figure 1). For 𝑋={1,3,4}, the 𝑋-components of 𝐻 are𝐾1={𝐴1,𝐴3} and the boundary of 𝑋 with 𝐾1 is {1,4},𝐾1={𝐴2} and the boundary of 𝑋 with 𝐾2 is {1,3},𝐾1={𝐴4,𝐴5} and the boundary of 𝑋 with 𝐾3 is {3,4}.

Theorem 3.4 (see [4]). Let 𝐻 be a connected hypergraph. A subset 𝑋 of 𝑉(𝐻) is 𝑚-convex if and only if either 𝑋=∅ or the boundary of 𝑋 with every 𝑋-component of 𝐻 is a clique of 𝐻.

As we noted above, in contrast with 𝑚-convexity in graphs, the original definition of 𝑚-convexity in hypergraphs was given without having recourse to any path type. We now prove that 𝑚-convexity in hypergraphs can be related to the following generalization of the notion of a chordless path in a graph, which (is different from that given in [15] and) is defined as follows.

A chord of a path 𝑝 is a pair of nonconsecutive vertices on 𝑝 which are adjacent in 𝐻. A path in 𝐻 is chordless if it has no chords.

Theorem 3.5. Let 𝐻 be a connected hypergraph. A subset 𝑋 of 𝑉(𝐻) is 𝑚-convex if and only if, for every chordless path 𝑝 joining two vertices in 𝑋, each vertex on 𝑝 belongs to 𝑋.

Proof. (if) Assume that 𝑋 contains 𝑉(𝑝) for every chordless path 𝑝 joining two vertices in 𝑋. By Theorem 3.4, it is sufficient to prove that the boundary of 𝑋 with every 𝑋-component of 𝐻 is a clique of 𝐻. Suppose, by contradiction, that there exists an 𝑋-component 𝐾 of 𝐻 such that the boundary of 𝑋 with 𝐾 contains two vertices 𝑢 and 𝑣 that are not adjacent in 𝐻. Let 𝑌 be the boundary of 𝑋 with 𝐾. Since 𝑢 and 𝑣 are not adjacent in 𝐻,𝑢 and 𝑣 are not adjacent in 𝐾 and, since 𝐾 is 𝑋-connected, there exists a 𝑢–𝑣 path 𝑝 in 𝐾 of length at least 2 such that 𝑌∩𝑉(𝑝)={𝑢,𝑣}. Let 𝑝 be a 𝑢–𝑣 path of minimum length in 𝐾(𝑝). Of course, 𝑉(𝑝)⊆𝑉(𝑝) and 𝑝is a chordless path; moreover, one has 𝑌∩𝑉(𝑝)={𝑢,𝑣}. Since 𝑢 and 𝑣 are not adjacent in 𝐾,𝑝must be of length at least 2 and, hence, there exists a vertex 𝑤 in 𝑉(𝑝)∩𝑉(𝐾) that does not belong to 𝑌. Since 𝑋∩𝑉(𝑝)=𝑌∩𝑉(𝑝)={𝑢,𝑣}, 𝑤 does not belong to 𝑋 so that 𝑋 does not contain 𝑉(𝑝) (contradiction).(only if) Assume that 𝑋 is 𝑚-convex. By Theorem 3.4, the boundary of 𝑋 with every 𝑋-component of 𝐻 is a clique of 𝐻. Suppose, by contradiction, that there exist two vertices 𝑢 and 𝑣 in 𝑋 and a chordless 𝑢–𝑣 path 𝑝=(𝑎0,𝐴1,𝑎1,…,𝐴𝑘,𝑎𝑘) such that 𝑉(𝑝)⧵𝑋≠∅. Let 𝑖=min{ℎ>1∶𝑎ℎ∉𝑋} and let 𝑗=min{ℎ>𝑖∶𝑎ℎ∈𝑋}. Of course, both 𝑎𝑖−1 and 𝑎𝑗 belong to the boundary of 𝑋 with the 𝑋-component of 𝐻 containing 𝑎𝑖 and, since the boundary of 𝑋 with every 𝑋-component of 𝐻 is a clique, 𝑎𝑖−1 and 𝑎𝑗 are adjacent. Since 𝑎𝑖−1 and 𝑎𝑗 are nonconsecutive on 𝑝, the pair {𝑎𝑖, 𝑎𝑗} is a chord of 𝑝 (contradiction).

Finally, let 𝐻 be a connected hypergraph with 𝑛 vertices and 𝑚 edges. An algorithm for computing 𝑚-convex hulls in a hypergraph 𝐻 is the “monophonic-closure algorithm” [4] which runs in 𝑂(𝑚𝑛) time if the prime hypergraph of 𝐻[2] is given. Since the time needed to construct the prime hypergraph of 𝐻[2] is 𝑂(𝑒𝑛) [21], where 𝑒 is the number of edges of 𝐻[2], and since 𝑒=𝑂(𝑛2), the time complexity of the monophonic-closure algorithm is 𝑂(𝑚𝑛+𝑛3). On the other hand, it is easy to check that (𝑎0,𝐴1,𝑎1,…,𝐴𝑘,𝑎𝑘) is a chordless path in 𝐻 if and only if (𝑎0,𝑎1,…,𝑎𝑘) is a chordless path in 𝐻[2], which implies that 𝑚(𝐻)=𝑚(𝐻[2]) so that, given 𝐻[2], the 𝑚-convex hull of any vertex set in 𝐻 can be computed in 𝑂(𝑛𝑒) time, that is, in 𝑂(𝑛3) time using the Kannan-Changat algorithm.

3.2.2. 𝑐-Convexity

A subset 𝑋 of 𝑉(𝐻) is 𝑐-convex if the boundary of 𝑋 with every 𝑋-component of 𝐻 is a partial edge of 𝐻. Let 𝐻 be a connected hypergraph with 𝑛 vertices and 𝑚 edges. It is proven in [4] that 𝑐-convex hulls can be computed using the Maier-Ullman algorithm [28], which runs in 𝑂(𝑚4𝑛) time. A more efficient algorithm is the “canonical-closure algorithm” [4], which runs in 𝑂(𝑚𝑛) time if the compact hypergraph of 𝐻 is given. Since the time needed to construct the compact hypergraph of 𝐻 is 𝑂(𝑚3𝑛) [23], the time complexity of the canonical-closure algorithm is 𝑂(𝑚3𝑛). Note that if 𝐻 is 𝛼-acyclic, then the time complexity of the algorithm reduces to 𝑂(𝑚𝑛); however, we can do better using the “selective-reduction algorithm” [18] which is linear in the size of 𝐻.

3.2.3. sp-Convexity

A path 𝑝 in 𝐻 is simple [3] if every two nonconsecutive vertices on 𝑝 are not adjacent in the partial hypergraph 𝐻(𝑝); equivalently, a path 𝑝 in 𝐻 is simple if 𝑝 is a chordless path in 𝐻(𝑝). A subset 𝑋 of 𝑉(𝐻) is sp-convex if, for every simple path 𝑝 joining two vertices in 𝑋, each vertex on 𝑝 belongs to 𝑋. Let 𝐻 be a connected hypergraph with 𝑛 vertices and 𝑚 edges. An efficient algorithm for computing sp-convex hulls was given in [29], which runs in 𝑂(𝑚𝑛3𝑒), where 𝑒 is the number of edges of the incidence graph of 𝐻. Since 𝑒≤𝑚𝑛, the time complexity of the algorithm is 𝑂(𝑚2𝑛4). However, if 𝐻 is 𝛽-acyclic, then using the Anstee-Farber algorithm [15], the sp-convex hull of vertex set 𝑋 can be computed in 𝑂(𝑚𝑛2) time simply by deleting the “nest” vertices of 𝐻 that are not in 𝑋 (see [12]).

3.2.4. ap-Convexity

Let 𝐻 be a connected hypergraph. The convexity space ap(𝐻) is defined in the same way as in Section 3.1, that is, a subset 𝑋 of 𝑉(𝐻) is ap-convex if, for every path 𝑝 joining two vertices in 𝑋, each vertex on 𝑝 belongs to 𝑋. Again one always has ap(𝐻)⊆sp(𝐻). In Section 6 we will give an efficient algorithm for computing ap-convex hulls.

4. 𝑐-Convexity versus 𝑚-Convexity

Since every partial edge is a clique, one always has 𝑐(𝐻)⊆𝑚(𝐻). In this section we characterize the class of hypergraphs 𝐻 for which 𝑐(𝐻)=𝑚(𝐻). First of all, observe that if 𝐻 is conformal, then every clique is a partial edge so that, by Theorem 3.4, every 𝑚-convex set of 𝐻 is also 𝑐-convex so that 𝑐(𝐻)=𝑚(𝐻). We will see that conformality is not a necessary condition for 𝑐(𝐻)=𝑚(𝐻).

A clique 𝑋 of 𝐻 is a boundary clique if there exists an 𝑋-component 𝐾 of 𝐻 such that 𝑋 equals the boundary of 𝑋 with 𝐾. A hypergraph 𝐻 is weakly conformal if every boundary-clique of 𝐻 is a partial edge of 𝐻. Of course, every conformal hypergraph is weakly conformal. The following is an example of a weakly conformal hypergraph that is not conformal.

Example 4.1. Consider the (hyper)graph 𝐻 of Figure 4. The only cliques of 𝐻 that are not boundary cliques are the two cliques with cardinality 3, namely, the sets {2,3,6} and {2,5,6}. Since each clique of 𝐻 with cardinality less than 3 is a partial edge of 𝐻, each boundary clique is a partial edge and, hence, 𝐻 is weakly conformal.

Figure 4

Theorem 4.2. Let 𝐻 be a connected hypergraph. The equality 𝑐(𝐻)=𝑚(𝐻) holds if and only if 𝐻 is weakly conformal.

Proof. (if) Let 𝑋 be any 𝑚-convex set. Let 𝐾 be any 𝑋-component of 𝐻, and let 𝑌 be the boundary of 𝑋 with 𝐾. By the very definition of 𝑚-convexity, 𝑌 is a clique; moreover, 𝐾 is also a 𝑌-component of 𝐻 and the boundary of 𝑌 with 𝐾 is 𝑌 itself. Therefore, 𝑌 is a boundary clique of 𝐻. Since 𝐻 is weakly conformal, 𝑌 is a partial edge of 𝐻. It follows that the boundary of 𝑋 with every 𝑋-component of 𝐻 is a partial edge of 𝐻 and, hence, 𝑋 is 𝑐-convex.(only if) Let 𝑋 be any boundary clique of 𝐻. Since 𝑋 is a clique, from the very definition of 𝑚-convexity it follows that 𝑋 is 𝑚-convex and, since 𝑐(𝐻)=𝑚(𝐻) by hypothesis, 𝑋 is 𝑐-convex. From the very definition of 𝑐-convexity, it follows that the boundary of 𝑋 with every 𝑋-component of 𝐻 is a partial edge. Finally, since 𝑋 is a boundary clique of 𝐻, there exists an 𝑋-component 𝐾 of 𝐻 such that 𝑋 is the boundary of 𝑋 with 𝐾. So, 𝑋 is a partial edge of 𝐻. It follows that 𝐻 is weakly conformal.

5. sp-Convexity versus 𝑐-Convexity and 𝑚-Convexity

In this section we characterize the class of hypergraphs 𝐻 for which sp(𝐻)=𝑐(𝐻) and the class of hypergraphs 𝐻 for which sp(𝐻)=𝑚(𝐻).

5.1. Equivalence between sp-Convexity and 𝑐-Convexity

Let 𝐻 be a connected hypergraph. We first prove that sp(𝐻)⊆𝑐(𝐻). To achieve this, we need the following two technical lemmas.

Lemma 5.1. If 𝑝 is a 𝑢–𝑣 path in 𝐻 and 𝑢 and 𝑣 are not adjacent in 𝐻(𝑝), then there exists in 𝐻(𝑝) a simple 𝑢–𝑣 path 𝑝 of length at least 2 and with 𝑉(𝑝)⊆𝑉(𝑝).

Proof. Let (𝑎0,𝐴1,𝑎1,…,𝐴𝑘,𝑎𝑘) be a 𝑢–𝑣 path. Let 𝑖(1)= max {ℎ∶𝑎ℎ and 𝑎0 are adjacent in 𝐻(𝑝)} and let 𝐴𝑗(1) be an edge of 𝐻(𝑝) that contains both 𝑎𝑖(1) and 𝑎0. Since 𝑢 and 𝑣 are not adjacent in 𝐻(𝑝), one has 𝑖(1)<𝑘. Consider the 𝑢–𝑣 path (𝑎0,𝐴𝑗(1),𝑎𝑖(1),…,𝐴𝑘,𝑎𝑘). Let 𝑖(2)=max{ℎ>𝑖(1)∶𝑎ℎ and 𝑎𝑖(1) are adjacent in 𝐻(𝑝)}, and let 𝐴𝑗(1) be any edge of 𝐻(𝑝) that contains both 𝑎𝑖(2) and 𝑎𝑖(1). If 𝑖(2)=𝑘, then the 𝑢–𝑣 path (𝑎0,𝐴𝑗(1),𝑎𝑖(1),𝐴𝑗(2),𝑎𝑘) is simple since 𝑎𝑘∉𝐴𝑗(1) and 𝑎0∉𝐴𝑗(2). If 𝑖(2)<𝑘,then let 𝑖(3)=max{ℎ>𝑖(2)∶𝑎ℎ and 𝑎𝑖(2) are adjacent in 𝐻(𝑝)}, and let 𝐴𝑗(3) be any edge of 𝐻(𝑝) that contains both 𝑎𝑖(3) are 𝑎𝑖(2). If 𝑖(3)=𝑘, then the 𝑢–𝑣 path (𝑎0,𝐴𝑗(1),𝑎𝑖(1),𝐴𝑗(2),𝑎𝑖(2),𝐴𝑗(3),𝑎𝑘) is simple since 𝑎𝑘∉𝐴𝑗(2) and 𝑎𝑖(1)∉𝐴𝑗(3), and so on.

The next lemma characterizes sp-convex sets.

Lemma 5.2. A subset 𝑋 of 𝑉(𝐻) is sp-convex if and only if either |𝑋|≤1 or, for every two distinct vertices 𝑢 and 𝑣 in 𝑋, there exists no 𝑋–𝑋 path joining 𝑢 and 𝑣 in the partial hypergraph 𝐻𝑢𝑣 of 𝐻 obtained by deleting the edges that contain both 𝑢 and 𝑣.

Proof. (only if) Assume that 𝑋 is sp-convex and |𝑋|>1. Suppose, by contradiction, that there exist two vertices 𝑢 and 𝑣 in 𝑋 and an 𝑋–𝑋 path joining 𝑢 and 𝑣 in 𝐻𝑢𝑣. By construction of 𝐻𝑢𝑣, the vertices 𝑢 and 𝑣 are not adjacent in 𝐻𝑢𝑣 so that by Lemma 5.1, there exists a simple 𝑋–𝑋 path 𝑝 of length at least 2 joining 𝑢 and 𝑣 in 𝐻𝑢𝑣. Since 𝑝 is also a simple path in 𝐻 and 𝑉(𝑝) is not contained in 𝑋,𝑋 is not sp-convex (contradiction).(if) If |𝑋|≤1 then 𝑋 is trivially sp-convex. Assume that |𝑋|>1 and, for every two distinct vertices 𝑢 and 𝑣 in 𝑋, there exists no 𝑋–𝑋 path joining 𝑢 and 𝑣 in 𝐻𝑢𝑣. Suppose, by contradiction, that 𝑋 is not sp-convex. Then, there exist two vertices 𝑢 and 𝑣 in 𝑋 and a simple 𝑢–𝑣 path 𝑝 in 𝐻 with 𝑉(𝑝)⧵𝑋≠∅. Let 𝑤 be a vertex in 𝑉(𝑝)⧵𝑋, and let 𝑝be the subpath of 𝑝 such that 𝑤 is on 𝑝and 𝑝is an 𝑋–𝑋 path. Of course,𝑝is a simple path in 𝐻 and has length at least 2. Let 𝑥 and 𝑦 be the vertices (in 𝑋) that are joined by 𝑝. Since 𝑝is a simple path in 𝐻, no edge of 𝐻(𝑝) contains both 𝑥 and 𝑦, so that 𝑝 is also a path in 𝐻𝑥𝑦. To sum up, 𝑝is an 𝑋–𝑋 path that in 𝐻𝑥𝑦 joins the vertices 𝑥 and 𝑦 in 𝑋 so that 𝑋 is not sp-convex (contradiction).

Example 5.3. Let 𝐻={𝐴1,𝐴2,𝐴3,𝐴4,𝐴5}, where 𝐴1={1}, 𝐴2={1,2}, 𝐴3={1,2,3},𝐴4={3}, 𝐴5={3,4}, and 𝐴6={3,4,5}. The hypergraph 𝐻 is shown in Figure 5. Let 𝑋={1,3,4}. Consider the three vertex pairs in 𝑋. For the vertex pair {1,3}, the partial hypergraph 𝐻13 of 𝐻 is {𝐴1,𝐴2,𝐴4,𝐴5,𝐴6} and, since 1 and 3 belong to distinct connected components of 𝐻13, there exists no 𝑋–𝑋 path joining 1 and 3 in 𝐻13. For the vertex pair {1,4}, the partial hypergraph 𝐻14 of 𝐻 is 𝐻 itself and, since every path joining 1 and 4 in 𝐻14 passes through 3, there exists no 𝑋–𝑋 path joining 1 and 4 in 𝐻14. For the vertex pair {3,4}, the partial hypergraph 𝐻34 of 𝐻 is {𝐴1,𝐴2,𝐴3,𝐴4} and, since 4 is not a vertex of 𝐻14, there exists no 𝑋–𝑋 path joining 3 and 4 in 𝐻34. By Lemma 5.2, the set 𝑋 is sp-convex, which is confirmed by the fact that the only simple paths joining two vertices in 𝑋 are (1,𝐴3,3), (1,𝐴3,3,𝐴5,4), (1,𝐴3,3,𝐴6,4), (3,𝐴5,4), and (3,𝐴6,4).

Figure 5

We first prove the inclusion sp(𝐻)⊆𝑐(𝐻).

Theorem 5.4. Let 𝐻 be a connected hypergraph. Every sp-convex set of 𝐻 is 𝑐-convex.

Proof. Suppose, by contradiction, that there exists an sp-convex set 𝑋 of 𝐻 that is not 𝑐-convex. Then, there exists an 𝑋-component 𝐾 of 𝐻 such that the boundary of 𝑋 with 𝐾 (i.e., the set 𝑋∩𝑉(𝐾)) is not a partial edge of 𝐻. Of course, |𝑋∩𝑉(𝐾)|≥2 and the boundary of 𝑋 with 𝐾 is not a partial edge of 𝐾. Let 𝐴 be an edge of 𝐾 such that for every edge 𝐴 of 𝐾, either 𝑋∩𝐴⊆𝐴 or there exists 𝑢∈(𝑋∩𝐴)⧵𝐴. Since the boundary of 𝑋 with 𝐾 is not a partial edge of 𝐾,𝑋∩𝐴is a proper subset of 𝑋∩𝑉(𝐾); therefore, there exist two vertices 𝑢0 and 𝑣 such that 𝑢0∈𝑋∩𝐴and 𝑣∈(𝑋∩𝑉(𝐾))⧵𝐴. Let 𝐴 be any edge of 𝐾 that contains 𝑣. Since 𝐾 is an 𝑋-component of 𝐻,𝐴≡𝑋𝐴, and, hence, there exists an 𝑋–𝑋 path 𝑝0=(𝑢0,𝐴,𝑣1,…,𝐴𝑘,𝑣) in 𝐾 of length at least 2 (i.e., 𝑘≥2) with 𝐴𝑘=𝐴. If no edge of 𝐻(𝑝0) contains both 𝑢0 and 𝑣, then 𝑋 is not sp-convex by Lemma 5.2 and a contradiction arises. Otherwise, let 𝐴ℎ(1) be the first edge on 𝑝0 that contains both 𝑢0 and 𝑣. Since 𝑣∈𝑋∩𝐴ℎ(1) and 𝑣∉𝐴, 𝑋∩𝐴ℎ(1) is not a subset of 𝐴so that there exists 𝑢1∈(𝑋∩𝐴)⧵𝐴ℎ(1). Consider the 𝑋–𝑋 path 𝑝1=(𝑢1,𝐴,𝑣1,…,𝐴ℎ(1),𝑣). If no edge of 𝐻(𝑝1) contains both 𝑢1 and 𝑣, then 𝑋 is not sp-convex by Lemma 5.2 and a contradiction arises. Otherwise, let 𝐴ℎ(2) be the first edge on 𝑝1 that contains both 𝑢1 and 𝑣. Since 𝑣∈𝑋∩𝐴ℎ(2) and 𝑣∉𝐴,𝑋∩𝐴ℎ(2) is not a subset of 𝐴 so that there exists 𝑢2∈(𝑋∩𝐴)⧵𝐴ℎ(2). Consider the 𝑋–𝑋 path 𝑝2=(𝑢2,𝐴,𝑤0,…,𝐴ℎ(2),𝑣). If no edge of 𝐻(𝑝2) contains both 𝑢2 and 𝑣, then 𝑋 is not sp-convex by Lemma 5.2 and a contradiction arises, and so on. Thus, ultimately one obtains an 𝑋–𝑋 path 𝑝 of length at least 2 that joins two vertices 𝑢 and 𝑣 in 𝑋 and is such that no edge of 𝐻(𝑝) contains both 𝑢 and 𝑣. By Lemma 5.2, 𝑋 is not sp-convex and a contradiction arises.

We now characterize the class of hypergraphs 𝐻 for which sp(𝐻)=𝑐(𝐻).

Theorem 5.5. Let 𝐻 be a connected hypergraph. The equality sp(𝐻)=𝑐(𝐻) holds if and only if 𝐻 is 𝛾-acyclic.

Proof. (if) Assume that 𝐻 is 𝛾-acyclic and suppose, by contradiction, that sp(𝐻)≠𝑐(𝐻). Let 𝑋 be a 𝑐-convex set that is not sp-convex. Then, there exist two vertices 𝑢 and 𝑣 in 𝑋 and a simple 𝑢–𝑣 path 𝑝 (of length at least 2) such that 𝑉(𝑝)⧵𝑋≠∅. Let 𝑝=(𝑎0,𝐴1,𝑎1,𝐴2,…,𝑎𝑘−1,𝐴𝑘,𝑎𝑘), let ℎ be such that 𝑎ℎ∉𝑋, let 𝑖=max{𝑙∶0≤𝑙<ℎ and 𝑎𝑙∈𝑋}, and let 𝑗=min{𝑙∶ℎ<𝑙≤𝑘 and 𝑎𝑙∈𝑋). Consider the subpath 𝑝=(𝑎𝑖,𝐴𝑖+1,𝑎𝑖+1,…,𝐴𝑗,𝑎𝑗) of 𝑝. Since 𝑝 is a simple path, 𝑝 is a simple path of length at least 2, and 𝑉(𝑝)∩𝑋={𝑎𝑖,𝑎𝑗} by construction. Let 𝐾 be the 𝑋-component of 𝐻 such that 𝐻(𝑝)⊆𝐾. Then, both 𝑎𝑖 and 𝑎𝑗 belong to the boundary of 𝑋 with 𝐾. Since 𝑋 is 𝑐-convex, the boundary of 𝑋 with 𝐾 is contained in an edge of 𝐻, say 𝐴. Since 𝑝is a simple path of length at least 2, and 𝐴 contains both 𝑎𝑖 and 𝑎𝑗, 𝐴 is not an edge of 𝐻(𝑝) so that 𝑐=(𝑎𝑖,𝐴𝑖+1,𝑎𝑖+1,…,𝐴𝑗,𝑎𝑗,𝐴,𝑎𝑖) is a cycle; moreover, since 𝑝is a path of length at least 2, 𝑐 has length at least 3. Distinguish two cases depending on whether or not 𝑉(𝑝)⊆𝐴.Case 1 (𝑉(𝑝)⊆𝐴). Then, 𝑐=(𝑎𝑖,𝐴𝑖+1,𝑎𝑖+1,𝐴𝑖+2,𝑎𝑖+2,𝐴,𝑎𝑖) is a cycle of length 3 and, since only the vertex 𝑎𝑖+1 belongs to the three edges of 𝑐,𝑐 is a 𝛾-cycle (contradiction).Case 2 (𝑉(𝑝)⧵𝐴≠∅). Then, there exists in 𝑉(𝑝) a vertex 𝑎𝑟∉𝐴 for some 𝑟, 𝑖<𝑟<𝑗. Let 𝑖∗=max{𝑙∶𝑖≤𝑙<𝑟 and 𝑎𝑙∈𝐴} and 𝑗∗=min{𝑙∶𝑟<𝑙≤𝑗 and 𝑎𝑙∈𝐴}. Then 𝑐=(𝑎𝑖∗,𝐴𝑖∗+1,𝑎𝑖∗+1,…,𝑎𝑗∗−1,𝐴𝑗∗,𝑎𝑗∗,𝐴,𝑎𝑖∗) is a cycle of length at least 3 and, since every vertex in 𝑉(𝑐) belongs to exactly two edges of 𝐻(𝑐), 𝑐is a 𝛾-cycle (contradiction).(only if) Assume that every 𝑐-convex set of 𝐻 is also sp-convex and suppose, by contradiction, that 𝐻 is not 𝛾-acyclic. Let 𝑐=(𝑎0,𝐴1,𝑎1,𝐴2,…,𝑎𝑘−1,𝐴𝑘,𝑎0), 𝑘≥3, be a 𝛾-cycle. Distinguish two cases depending on whether or not each vertex in 𝑉(𝑐) belongs to exactly two edges of 𝐻(𝑐).Case 1. Each vertex in 𝑉(𝑐) belongs to exactly two edges of 𝐻(𝑐). Thus, 𝑐 is a 𝛾-cycle. Let 𝑋={𝑎1,𝑎2}. Since 𝑋⊆𝐴2, for each 𝑋-component 𝐾 of 𝐻,𝑋∩𝑉(𝐾) is a partial edge of 𝐻 and, hence, 𝑋 is 𝑐-convex. On the other hand, since each vertex in 𝑉(𝑐) belongs to exactly two edges of 𝐻(𝑐), (𝑎2,…,𝐴𝑘,𝑎0,𝐴1,𝑎1) is simple path of length at least 2 and, since 𝑎0∉𝑋,𝑋 is not sp-convex (contradiction).Case 2. There exists a vertex in 𝑉(𝑐) that belongs to more than two edges of 𝐻(𝑐). Without loss of generality, let it be 𝑎0. Since 𝑐 is a 𝛾-cycle, each 𝑎ℎ, ℎ≠0, belongs to exactly two edges of 𝐻(𝑐). Let 𝐴𝑖 be any edge of 𝐻(𝑐)⧵{𝐴1, 𝐴𝑘} containing 𝑎0, and let 𝑋={𝑎𝑖−1, 𝑎𝑖}. Since 𝑋⊆𝐴𝑖, for each 𝑋-component 𝐾 of 𝐻,𝑋∩𝑉(𝐾) is a partial edge of 𝐻 and, hence, 𝑋 is 𝑐-convex. Let
𝑟=maxℎ∶1≤ℎ<𝑖,𝑎0∈𝐴ℎ𝑠=minℎ∶𝑖<ℎ≤𝑘,𝑎0∈𝐴ℎ.(5.1)
It is easy to see that (𝑎𝑖,𝐴𝑖+1,𝑎𝑖+1,…,𝐴𝑠,𝑎0,𝐴𝑟,𝑎𝑟,…,𝐴𝑖−1,𝑎𝑖−1) is a simple path of length at least 2 and, since 𝑎0∉𝑋,𝑋 is not sp-convex (contradiction).

Let 𝐻 be a connected hypergraph with 𝑛 vertices and 𝑚 edges. If 𝐻 is 𝛾-acyclic, then 𝐻 is 𝛽-acyclic and, hence, sp-convex hulls can be computed in 𝑂(𝑚𝑛2) using the Anstee-Farber algorithm. On the other hand, if 𝐻 is 𝛾-acyclic, then 𝐻 is 𝛼-acyclic and, hence, 𝑐-convex hulls can be computed in linear time using the Tarjan-Yannakakis algorithm. By Theorem 5.5, sp-convex hulls can be computed in linear time, that is, more efficiently than using the Anstee-Farber algorithm.

5.2. Equivalence between sp-Convexity and 𝑚-Convexity

Note that, by Theorem 3.5 and by the fact that every chordless path in 𝐻 is a simple path in 𝐻, one always has sp(𝐻)⊆𝑚(𝐻). The following is another convexity-theoretic characterization of 𝛾-acyclic hypergraphs.

Theorem 5.6. A connected hypergraph 𝐻 is γ-acyclic if and only if sp(𝐻)=𝑚(𝐻).

Proof. (only if) By hypothesis, 𝐻 is 𝛾-acyclic so that sp(𝐻)=𝑐(𝐻) by Theorem 5.5. Moreover, by Proposition 2.4, 𝐻 is conformal and, hence, weakly conformal, so that 𝑐(𝐻)=𝑚(𝐻) by Theorem 4.2. To sum up, sp(𝐻)=𝑐(𝐻)=𝑚(𝐻).(if) By hypothesis, sp(𝐻)=𝑚(𝐻). Since sp(𝐻)⊆𝑐(𝐻) by Theorem 5.5 and 𝑐(𝐻)⊆𝑚(𝐻), one has sp(𝐻)=𝑐(𝐻) so that 𝐻 is 𝛾-acyclic again by Theorem 5.5.

By Theorem 5.6, 𝑚-convex hulls can be computed in linear time more efficiently than using the monophonic-closure algorithm.

6. ap-Convexity versus sp-Convexity, 𝑐-Convexity, and 𝑚-Convexity

Let 𝐻 be a connected hypergraph. In this section we characterize the three classes of hypergraphs 𝐻 for which ap(𝐻)=sp(𝐻), ap(𝐻)=𝑐(𝐻), and ap(𝐻)=𝑚(𝐻). To achieve this, we first give a polynomial algorithm for computing ap-convex hulls.

6.1. Computing ap-Convex Hulls

We represent 𝐻 by its incidence graph 𝐺(𝐻).

Remark 6.1. For every two vertices 𝑢 and 𝑣 of 𝐻, every 𝑢–𝑣 path in 𝐻 is a 𝑢–𝑣 path in 𝐺(𝐻) and vice versa; moreover, every cycle in 𝐻 is a cycle in 𝐺 and vice versa.To avoid ambiguities, we call the vertices and edges of 𝐺(𝐻) the nodes and arcs of 𝐺(𝐻), respectively. A node 𝑣 of 𝐺(𝐻) is a vertex-node or an edge-node depending on whether 𝑣∈𝑉(𝐻) or 𝑣∈𝐻. Moreover, we call cutpoints the cut vertices of 𝐺(𝐻); furthermore, a cutpoint of 𝐺(𝐻) is a vertex-cutpoint or an edge-cutpoint depending on whether it is a vertex-node or an edge-node. Note that, if 𝑎 is a vertex-cutpoint of 𝐺(𝐻), then either the induced subhypergraph 𝐻⧵{𝑎} is not connected (see the vertex-node 3 in Figure 6) or the singleton {𝑎} is an edge of 𝐻 (see the vertex-node 1 in Figure 6); moreover, if 𝐴 is an edge-cutpoint of 𝐺(𝐻), then either the partial hypergraph 𝐻⧵{𝐴} is not connected (see the edge-node 𝐴3 in Figure 6) or there exist one or more vertices of 𝐻 that belong to 𝐴 and to no other edge of 𝐻 (see the edge-node 𝐴6 in Figure 6). Our algorithm works with the “block-cutpoint tree” of 𝐺(𝐻), which is defined as follows. Let 𝑇 be the bipartite graph whose nodes are the cutpoints and blocks of 𝐺(𝐻) and where (𝑣,𝐵) is an arc if the cutpoint 𝑣 of 𝐺(𝐻) is a node of the block 𝐵 of 𝐺(𝐻). A node of 𝑇 is a block-node if it is a block of 𝐺(𝐻) and a cutpoint-node otherwise. It is well known [14] that 𝑇 is a tree, which is called the block cut-vertex tree of 𝐺(𝐻). We also label each block-node 𝐵 of 𝑇 by the vertex set 𝑉(𝐵).

Figure 6

Example 6.1. Consider again the hypergraph 𝐻={𝐴1,𝐴2,𝐴3,𝐴4,𝐴5,𝐴6} of Example 5.3 (see Figure 5). The incidence graph 𝐺(𝐻) of 𝐻 is shown in Figure 6, in which the cutpoints of 𝐺(𝐻) are circled.Note that the induced subhypergraph 𝐻⧵{1} is connected and the induced subhypergraph 𝐻⧵{3} is not connected; moreover, the partial hypergraph 𝐻⧵{𝐴3} is not connected and the partial hypergraph 𝐻⧵{𝐴5} is connected.The blocks of 𝐺(𝐻) are reported in Figure 7.The block-cutpoint tree 𝑇 of 𝐺(𝐻) is shown in Figure 8.The six block-nodes of 𝑇 are labeled as follows:
𝐵𝑏𝑙𝑜𝑐𝑘-𝑛𝑜𝑑𝑒𝑙𝑎𝑏𝑒𝑙1𝑉𝐵1𝐵={1}2𝑉𝐵2𝐵={1,2}3𝑉𝐵3𝐵={3}4𝑉𝐵4𝐵={3}5𝑉𝐵5=𝐵{3,4}6𝑉𝐵6={5}.(6.1)

Figure 7

Figure 8

Note that each leaf of 𝑇 is a block-node. Moreover, if 𝐻 is not a trivial hypergraph and 𝑇 is a one-point tree, then the node of 𝑇 is a nontrivial block of 𝐺(𝐻). Finally, if 𝐵 is a trivial block of 𝐺(𝐻) with 𝐻(𝐵)={𝐴} and 𝑉(𝐵)={𝑢}, then the block-node 𝐵 of 𝑇 is not a leaf if and only if 𝑢 and 𝐴 are both cutpoints of 𝐺(𝐻).

Algorithm 1 constructs the ap-convex hull (𝑌) of any subset 𝑋 of 𝑉(𝐻).

Algorithm 1

Example 6.1 (continued). When we apply Algorithm 1 with input 𝑋={1,3}, the tree 𝑇 resulting from the pruning of 𝑇 is shown in Figure 9. So, the output of Algorithm 1 is 𝑌={1,2,3}.When we apply Algorithm 1 with input 𝑋={1,4}, the tree 𝑇 resulting from the pruning of 𝑇 is shown in Figure 10. So, the output of Algorithm 1 is 𝑌={1,2,3,4}.

Figure 9

Figure 10

Remark 6.2. Each time a block-node 𝐵 is deleted during the pruning process, either 𝑋∩𝑉(𝐵)=∅ or |𝑋∩𝑉(𝐵)|=1 and the vertex in 𝑋∩𝑉(𝐵) belongs to 𝑉(𝐵) for some undeleted block-node 𝐵. Therefore, one has that 𝑋 is a subset of 𝑌.

Fact 1. Each leaf of 𝑇 is a block-node and, if 𝐵 is a leaf of 𝑇, then 𝑉(𝐵)∩𝑋≠∅. If|𝑋|>1 and 𝑇is a one-point tree, then the block-node of 𝑇is a nontrivial block of 𝐺(𝐻).

Fact 2. Let 𝐵 be a block-node of 𝑇 that is a trivial block of 𝐺(𝐻) with 𝐻(𝐵)={𝐴} and 𝑉(𝐵)={𝑢}. If|𝑋|>1, then 𝑇 is not a one-point tree and, furthermore,(i)if 𝑢∈𝑋 then 𝐴 is a node of 𝑇 adjacent to 𝐵 (see the cutpoint-node 𝐴3 and the block- node 𝐵3 in Figure 7) for, otherwise, the block-node 𝐵 would be a leaf of 𝑇 and 𝑢 would be the only node of 𝑇adjacent to 𝐵 and, since 𝑉(𝐵)∩𝑋={𝑢}, the leaf 𝐵 of 𝑇would have been deleted at step (1),(ii)if 𝑢∉𝑋 then 𝐵 is not a leaf of 𝑇(see the block-node 𝐵3 in Figure 10) for, otherwise, since 𝑉(𝐵)∩𝑋=∅, the block-node 𝐵 would have been deleted at step (1).

Fact 3. If a cutpoint-node of 𝑇is a vertex-node of 𝐺(𝐻), say 𝑢 (see the cutpoint-node 3 in Figure 8), then there exist two leaves 𝐵 and 𝐵of 𝑇 such that every 𝐵–𝐵path in 𝑇passes through the node 𝑢. Furthermore, since 𝑉(𝐵)∩𝑋≠∅ (by Fact 1), there exists a vertex 𝑣≠𝑢 in 𝑋∩𝑉(𝐵) for, otherwise, 𝑋∩𝑉(𝐵)={𝑢} and, since 𝑢 is the node adjacent to the leaf 𝐵, the leaf 𝐵 of 𝑇 would have been deleted at step 1. Analogously, there exists a vertex 𝑣≠𝑢 in 𝑋∩𝑉(𝐵). Finally, every 𝑣–𝑣path in 𝐺(𝐻) passes through 𝑢.

Theorem 6.3. Let 𝐻 be a connected hypergraph and let 𝑋 be a subset of 𝑉(𝐻). Algorithm 1 correctly computes the ap-convex hull of 𝑋.

Proof. Let 𝑌 be the output of Algorithm 1 and let ⟨𝑋⟩ap denote the ap-convex hull of 𝑋. If |𝑋|≤1, then 𝑌=𝑋 (see step (2)) and ⟨𝑋⟩ap=𝑋, which proves the statement. Assume that |𝑋|>1. We first prove that ⟨𝑋⟩ap⊆𝑌 and, then, 𝑌⊆⟨𝑋⟩ap.Proof of ⟨𝑋⟩ap⊆𝑌. By Remark 6.2, it is sufficient to prove that 𝑌 is ap-convex. If 𝑌=𝑉(𝐻) then the statement is trivially true. Assume that 𝑌≠𝑉(𝐻) and let 𝑢 be any vertex in 𝑉(𝐻)⧵𝑌. Since 𝑢∉𝑌, 𝑢 belongs to 𝑉(𝐵) for some block-node 𝐵 of 𝑇 that has been deleted during the pruning process. Let 𝐵∗ be the last of such block-nodes and let 𝑛 be the cutpoint-node adjacent to 𝐵∗ that is in every path joining 𝐵∗ and every node of the tree 𝑇 resulting from the pruning process. Note that if 𝑛 is a vertex-cutpoint of 𝐺(𝐻) then, since 𝐵∗ is the block-node with 𝑢∈𝑉(𝐵∗) that is last deleted, one has 𝑛≠𝑢 even if 𝑢 is cutpoint-node of 𝑇 (see Figure 11). Therefore, every path in 𝐺(𝐻) joining 𝑢 to any vertex in 𝑌 must pass through 𝑛 and, hence, no path in 𝐺(𝐻) joining two vertices in 𝑌 can pass through 𝑢 and, by Remark 6.1, the same holds in 𝐻, which proves that 𝑌 is ap-convex.Proof of 𝑌⊆⟨𝑋⟩ap. Consider again the tree 𝑇 resulting from the pruning process. By Fact 1, each leaf of 𝑇 is a block-node. Let 𝐿 be the set of the leaves of 𝑇. Thus, for each 𝐵∈𝐿, |𝑋∩𝑉(𝐵)|≥1 and, if |𝑋∩𝑉(𝐵)|=1, then the vertex in 𝑋∩𝑉(𝐵) is not the cutpoint-node adjacent to 𝐵 in 𝑇. Let 𝑋⋃=𝑋∩(𝐵∈𝐿𝑉(𝐵)) and let ⟨𝑋⟩ap denote the ap-convex hull of 𝑋. We now prove that 𝑌⊆⟨𝑋⟩ap. Consider the subgraph 𝐺of 𝐺(𝐻) formed by the block-nodes of 𝑇. First of all, observe that any cutpoint of 𝐺lies on some path joining two vertex-nodes in 𝑋. It follows that, for every trivial block 𝐵, the vertex in 𝑉(𝐵) lies on some path joining two vertex-nodes in 𝑋; moreover, for every nontrivial block 𝐵 of 𝐺, there exist at least two nodes of 𝐵 that lie on some path joining two vertex-nodes in 𝑋 and, by Proposition 2.1, each node of 𝐵 lies on some path joining two vertex-nodes in 𝑋. Therefore, every vertex-node of 𝐺 lies on some path joining two vertex-nodes in 𝑋, that is, 𝑉(𝐺)⊆⟨𝑋⟩ap. By step (2) of Algorithm 1, 𝑉(𝐺)=𝑌 and, hence, 𝑌⊆⟨𝑋⟩ap. Finally, since 𝑋⊆𝑋, one has ⟨𝑋⟩ap⊆⟨𝑋⟩ap so that 𝑌⊆⟨𝑋⟩ap.

Figure 11

The following result, which is an immediate consequence of Theorem 6.3, will be used in the sequel.

Corollary 6.4. Let 𝐻 be a connected hypergraph. For every block 𝐵 of 𝐺(𝐻), the subsets of 𝑉(𝐵) that are ap-convex are precisely the empty set, the singletons and 𝑉(𝐵).

6.2. Equivalence between ap-Convexity and sp-Convexity

Recall from Section 3.2.4 that one always has ap(𝐻)⊆sp(𝐻). We will prove that ap(𝐻)=sp(𝐻) if and only if the incidence graph 𝐺(𝐻) of 𝐻 satisfies the following two conditions:(C1) every edge-cutpoint of 𝐺(𝐻) is a node of only trivial blocks of 𝐺(𝐻);(C2) for every nontrivial block 𝐵 of 𝐺(𝐻), and for every 𝑋⊆𝑉(𝐵) with 1<|𝑋|<|𝑉(𝐵)|, there exist two distinct vertex-nodes 𝑢 and 𝑣 in 𝑋 and an 𝑋–𝑋 path joining 𝑢 and 𝑣 in the induced subgraph 𝐵𝑢𝑣 of 𝐵 obtained by deleting the edge-nodes that are adjacent to both 𝑢 and 𝑣.

Note that every graph satisfies (C1) owing to the fact that every edge contains exactly two vertices.

Remark 6.5. If (C1) holds then, for every nontrivial block 𝐵 of 𝐺(𝐻), one has |𝑉(𝐵)|≥3 for, otherwise (i.e., if |𝑉(𝐵)|=2), 𝐵 would contain an edge-cutpoint of 𝐺(𝐻) (see Figure 12) owing to the fact that distinct edge-nodes of 𝐺(𝐻) have distinct sets of adjacent vertex-nodes.In order to characterize the class of hypergraphs 𝐻 for which ap(𝐻)=sp(𝐻), we first restate Lemma 5.2 as follows.

Figure 12

Lemma 6.6. Let 𝐺 be the incidence graph of 𝐻. A subset 𝑋 of 𝑉(𝐻) is sp-convex if and only if either |𝑋|≤1 or, for every two distinct vertex-nodes 𝑢 and 𝑣 in 𝑋, there exists no 𝑋–𝑋 path joining 𝑢 and 𝑣 in the induced graph 𝐺𝑢𝑣 of 𝐺(𝐻) obtained by deleting the edge-nodes adjacent to both 𝑢 and 𝑣.

Corollary 6.7. Let 𝐺 be the incidence graph of 𝐻. If condition (C2) holds then, for every block 𝐵 of 𝐺(𝐻), the subsets of 𝑉(𝐵) that are sp-convex are precisely the empty set, the singletons and 𝑉(𝐵).

Proof. The empty set and the singletons are trivially sp-convex. Moreover, by Lemma 6.6 and condition (C2), for every nontrivial block of 𝐺(𝐻), no subset 𝑋 of 𝑉(𝐵) with 1<|𝑋|<|𝑉(𝐵)| is sp-convex.

Theorem 6.8. Let 𝐻 be a connected graph. The equality ap(𝐻)=sp(𝐻) holds if and only if the incidence graph 𝐺(𝐻) of 𝐻 satisfies both conditions (C1) and (C2).

Proof. (only if) We first prove(i)if ap(𝐻)=sp(𝐻) then (C1)and, then,(ii)if ap(𝐻)=sp(𝐻) then (C2).Proof of (i). Suppose, by contradiction, that condition (C1) does not hold. Then, there exists an edge-cutpoint 𝐴 of 𝐺(𝐻) that is a node of a nontrivial block 𝐵 of 𝐺(𝐻). Let 𝑣∈𝐴∩𝑉(𝐵), let 𝑢∈𝐴⧵𝑉(𝐵), and let 𝐵 be the block containing both 𝐴 and 𝑢 (see Figure 13).By Theorem 6.3, the ap-convex hull of {𝑢,𝑣} is ⋃𝑉(𝐵)𝑉(𝐵), which is a proper superset of {𝑢,𝑣} as 𝐵 is a nontrivial block of 𝐺(𝐻). On the other hand, since (𝑢,𝐴,𝑣) is the only simple 𝑢–𝑣 path in 𝐻, the set {𝑢,𝑣} is sp-convex, which contradicts the hypothesis ap(𝐻)=sp(𝐻).Proof of (ii). Let 𝐵 be any nontrivial block of 𝐺(𝐻). Note that, since (C1) holds, by Remark 6.5 one has |𝑉(𝐵)|≥3. Let 𝑋 be any subset of 𝑉(𝐵) with 1<|𝑋|<|𝑉(𝐵)|. By Corollary 6.4, the set 𝑋 is not ap-convex and, since ap(𝐻)=sp(𝐻), 𝑋 is not sp-convex. By Lemma 6.6, there exist two distinct vertex-nodes 𝑢 and 𝑣 in 𝑋 and an 𝑋–𝑋 path joining 𝑢 and 𝑣 in 𝐺𝑢𝑣. Since 𝑋⊆𝑉(𝐵), by Proposition 2.2 every path joining 𝑢 and 𝑣 in 𝐺𝑢𝑣 is also a path in 𝐵𝑢𝑣; therefore, there exists an 𝑋–𝑋 path joining 𝑢 and 𝑣 in 𝐵𝑢𝑣, which proves that condition (C2) holds.(if) Let 𝑋 be any subset of 𝑉(𝐻) with |𝑋|>1, let ⟨𝑋⟩ap and ⟨𝑋⟩sp denote the ap-convex hull and the sp-convex hull of 𝑋, respectively. Of course, 𝑋⊆⟨𝑋⟩sp⊆⟨𝑋⟩ap. Therefore, it is sufficient to prove that ⟨𝑋⟩ap⊆⟨𝑋⟩sp. By Theorem 6.3, ⟨𝑋⟩ap is the output of Algorithm 1, that is, if 𝑇 is the tree resulting from the pruning of the block-cutpoint tree of 𝐺(𝐻), then ⟨𝑋⟩ap is the union of the sets 𝑉(𝐵) for all block-nodes 𝐵 of 𝑇. So, we need to prove that, for every block-node 𝐵 of 𝑇, one has 𝑉(𝐵)⊆⟨𝑋⟩sp. Distinguish two cases depending on whether or not 𝐵 is a trivial block of 𝐺(𝐻).Case 1. 𝐵 is a trivial block of 𝐺(𝐻). Let 𝑢 be the unique vertex in 𝑉(𝐵). If 𝑢∈𝑋 then 𝑢∈⟨𝑋⟩sp since 𝑋⊆⟨𝑋⟩sp. Consider now the case that 𝑢∉𝑋. Since 𝑋∩𝑉(𝐵)=∅, Fact 1, the block-node 𝐵 of 𝑇 is not a leaf of 𝑇. By Fact 2, 𝑢 is one of the nodes of 𝑇 adjacent to the block-node 𝐵. By Fact 3, there exist two vertices 𝑣 and 𝑣 in 𝑋 such that every 𝑣–𝑣 path in 𝐺(𝐻) passes through 𝑢. Since every 𝑣–𝑣path in 𝐻 is a 𝑣–𝑣path in 𝐺(𝐻) (see Remark 6.1), every simple 𝑣–𝑣path in 𝐻 passes through 𝑢 so that 𝑢 belongs to ⟨{𝑣,𝑣}⟩sp and, hence, to ⟨𝑋⟩sp, which proves that 𝑉(𝐵)⊆⟨𝑋⟩sp.Case 2. 𝐵 is a nontrivial block of 𝐺(𝐻). Consider first the case that